Del Pezzo surfaces and their automorphism groups play a key role in the classification up to conjugacy of finite subgroups of the Cremona group of the plane. Over an algebraically closed field, they are completely classified together with their automorphism groups. In this talk, we will focus on real rational Del Pezzo surfaces of degree 4. Unlike larger degrees, the degree 4 case involves an infinite moduli space of surfaces, already over the complex numbers. We will explain how studying the actions of automorphisms and of the Galois group on the conic bundle structures enables us to give a complete description of their automorphism groups by generators in terms of automorphisms and birational automorphisms.